# Solve for x(t) and v(t) given a Force equation using Mathematica

## Homework Equations

F= ma or F= md^2x/dt^2

## The Attempt at a Solution

I know that this second order differential is non linear. I attempted to solve the problem as -k/x^2 = md^2x/dt^2 but I'm getting trouble since it is a second order ODE and I haven't learned how to solve those yet. I was also looking at my class notes to a similar problem but my professor included potential energy? I'm a bit confused. Any help would be greatly appreciated. If you can also show me how to solve this on Mathematica, that would also be a plus. I don't have any experience with it.

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update: i think i may have solved the first part? Now I need to figure out to put in on Mathematica.

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ZapperZ
Staff Emeritus
You need to use the chain rule from calculus.

You know that a = dv/dt.

However, this can also be written as a = (dv/dx) * (dx/dt) = v (dv/dx).

This means that your force equation a = F/m can now be written as

$$v \frac{dv}{dx} = \frac{F}{m}$$

Consequently, what you need to solve (or integrate) is

$$v dv = \frac{F}{m} dx$$

since F is given as a function of x.

Zz.